Low-regularity strong cosmic censorship: generic inextendibility of the maximal globally hyperbolic development as a weak solution of the Einstein equations

For generic asymptotically flat (or compact) initial data for the Einstein equations, is the maximal globally hyperbolic development (MGHD) inextendible as a weak solution, i.e., as a Lorentzian manifold with a continuous metric and Christoffel symbols locally in $L^2$? This is Christodoulou's low-regularity formulation of Penrose's strong cosmic censorship conjecture, identified as open in the source paper (Galloway–Ling–Sbierski, 2017). The purely $C^0$-formulation is now known to be false (Kerr and two-ended charged Cauchy horizons are $C^0$-stable), which makes the weak-solution threshold the sharp deterministic boundary; the corresponding generic $L^2$-Christoffel inextendibility is proven only in spherical symmetry for a two-ended charged scalar model at the $C^2$ level, and conditionally near Kerr at the Lipschitz level. A companion open problem, likewise flagged by the source, asks for a systematic classification of the strength of metric breakdown (the minimal regularity any extension must lose), so far mapped only case-by-case.

Background

An initial value problem for the Einstein equations prescribes a Riemannian metric and second fundamental form (plus matter data) on a spacelike hypersurface; Choquet-Bruhat theory yields a unique maximal globally hyperbolic development (MGHD), the largest solution determined by that data in which causality is well-posed (a Lorentzian manifold is globally hyperbolic if it admits a Cauchy hypersurface met exactly once by every inextendible causal curve). A Lorentzian manifold is $C^k$-inextendible if it admits no isometric embedding as a proper subset of a larger Lorentzian manifold of the same dimension with $C^k$-regular metric; for $k \geq 2$ geodesic completeness gives a classical obstruction, but at the level of merely continuous metrics the classical arguments break down and special causal-theoretic tools are needed (Sämann's existence of length-maximizing causal curves for globally hyperbolic continuous metrics underlies the source paper's theorem that a timelike geodesically complete globally hyperbolic spacetime is $C^0$-inextendible). Low regularity matters because a solution of the Einstein equations in the weak sense requires only a continuous metric whose Christoffel symbols (the metric's first derivatives, which combine into the connection) are locally square-integrable in some coordinate system; ruling out an extension of this regularity is exactly what rules out continuing the solution as a weak solution. Penrose's strong cosmic censorship conjecture asserts, roughly, that the MGHD of generic initial data is the whole physical spacetime, so that general relativity is generically deterministic; Christodoulou's precise version (prologue of his 2009 monograph) states that for generic asymptotically flat (or compact) initial data the MGHD is inextendible as a weak solution. The question is sharp rather than formal: the inner (Cauchy) horizons of the charged Reissner–Nordström and rotating Kerr exact black holes are null boundaries across which the metric does extend continuously, and Dafermos (2003, spherical symmetry) and Dafermos–Luk (2017, without symmetry, conditional on exterior stability of Kerr) proved such Cauchy horizons are $C^0$-stable under perturbation, refuting the $C^0$-version of the conjecture; however, the same dynamics exhibits mass inflation (blow-up of a renormalized Hawking mass at the horizon) and curvature blow-up suggesting the horizon is a weak null singularity through which no weak solution can continue. Galloway–Ling–Sbierski (2017) therefore identify as the central open problems (a) the generic weak-solution inextendibility statement itself and (b) a classification of the strength of the breakdown of the metric — the minimal regularity that must be lost in any extension, i.e., a taxonomy of singularity strength (Milne-like Friedmann–Lemaître–Robertson–Walker cosmologies are $C^0$-extendible, Schwarzschild is $C^0$-inextendible, Kerr interiors are $C^0$-extendible but expected Lipschitz-inextendible).

Problem Statement

Prove or refute Christodoulou's low-regularity strong cosmic censorship conjecture: for generic asymptotically flat (or compact) initial data for the Einstein equations, in a precisely specified genericity class (e.g., open and dense, or residual, in a standard function-space topology on the data), the maximal globally hyperbolic development is inextendible as a weak solution — that is, it admits no isometric embedding as a proper subset of a Lorentzian manifold whose metric is continuous with Christoffel symbols locally square-integrable. A refutation consists of exhibiting a class of such data that is generic in the same sense fixed for the conjecture (open and dense, or residual) whose maximal globally hyperbolic developments nevertheless admit extensions of exactly this regularity. A separate companion open question, stated by the same source, asks for a classification of the strength of the breakdown of the metric across the boundary of the maximal globally hyperbolic development — for each class of data, determine the minimal regularity that any putative extension must lose, thereby giving a quantitative taxonomy of singularity strength for solutions of the Einstein equations.

References

1. Gregory J. Galloway, Eric Ling, Jan Sbierski, Timelike completeness as an obstruction to $C^0$-extensions, Communications in Mathematical Physics (2017), DOI 10.1007/s00220-017-3019-2, arXiv:1704.00353, https://arxiv.org/abs/1704.00353 (source paper)
2. Demetrios Christodoulou, The Formation of Black Holes in General Relativity, European Mathematical Society (2009), DOI 10.4171/068, https://doi.org/10.4171/068 (prologue containing the weak-solution formulation of strong cosmic censorship cited by the source paper)
3. Mihalis Dafermos, Stability and instability of the Cauchy horizon for the spherically symmetric Einstein–Maxwell-scalar field equations, Annals of Mathematics 158 (2003), DOI 10.4007/annals.2003.158.875, https://doi.org/10.4007/annals.2003.158.875
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